Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gsumhashmul Structured version   Visualization version   GIF version

Theorem gsumhashmul 31516
Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumhashmul.b 𝐵 = (Base‘𝐺)
gsumhashmul.z 0 = (0g𝐺)
gsumhashmul.x · = (.g𝐺)
gsumhashmul.g (𝜑𝐺 ∈ CMnd)
gsumhashmul.f (𝜑𝐹:𝐴𝐵)
gsumhashmul.1 (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumhashmul (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥
Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul
Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumhashmul.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
2 suppssdm 8055 . . . . . . . 8 (𝐹 supp 0 ) ⊆ dom 𝐹
32, 1fssdm 6665 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
41, 3feqresmpt 6888 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥)))
54oveq2d 7345 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))))
6 gsumhashmul.b . . . . . 6 𝐵 = (Base‘𝐺)
7 gsumhashmul.z . . . . . 6 0 = (0g𝐺)
8 gsumhashmul.g . . . . . 6 (𝜑𝐺 ∈ CMnd)
9 gsumhashmul.1 . . . . . . . 8 (𝜑𝐹 finSupp 0 )
10 relfsupp 9220 . . . . . . . . 9 Rel finSupp
1110brrelex1i 5668 . . . . . . . 8 (𝐹 finSupp 0𝐹 ∈ V)
129, 11syl 17 . . . . . . 7 (𝜑𝐹 ∈ V)
131ffnd 6646 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
1412, 13fndmexd 7813 . . . . . 6 (𝜑𝐴 ∈ V)
15 ssidd 3954 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
166, 7, 8, 14, 1, 15, 9gsumres 19601 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17 nfcv 2904 . . . . . 6 𝑥(𝐹‘(1st𝑧))
18 fveq2 6819 . . . . . 6 (𝑥 = (1st𝑧) → (𝐹𝑥) = (𝐹‘(1st𝑧)))
199fsuppimpd 9225 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20 ssidd 3954 . . . . . 6 (𝜑𝐵𝐵)
211adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴𝐵)
223sselda 3931 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥𝐴)
2321, 22ffvelcdmd 7012 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹𝑥) ∈ 𝐵)
241ffund 6649 . . . . . . . . 9 (𝜑 → Fun 𝐹)
25 funrel 6495 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
26 reldif 5751 . . . . . . . . 9 (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
2724, 25, 263syl 18 . . . . . . . 8 (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28 1stdm 7941 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
2927, 28sylan 580 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
307fvexi 6833 . . . . . . . . . . . 12 0 ∈ V
3130a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
32 fressupp 31222 . . . . . . . . . . 11 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3324, 12, 31, 32syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3433dmeqd 5841 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
352a1i 11 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36 ssdmres 5940 . . . . . . . . . 10 ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3735, 36sylib 217 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3834, 37eqtr3d 2778 . . . . . . . 8 (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
3938adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
4029, 39eleqtrd 2839 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ (𝐹 supp 0 ))
4124funresd 6521 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
4241adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
4337eleq2d 2822 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
4443biimpar 478 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
4645fvresd 6839 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
47 funopfvb 6875 . . . . . . . . . . 11 ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
4847biimpa 477 . . . . . . . . . 10 (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥)) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
4942, 44, 46, 48syl21anc 835 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
5033adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
5149, 50eleqtrd 2839 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52 eqeq2 2748 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (𝑧 = 𝑣𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
5352bibi2d 342 . . . . . . . . . 10 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → ((𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5453ralbidv 3170 . . . . . . . . 9 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5554adantl 482 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
56 fvexd 6834 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (2nd𝑧) ∈ V)
5727ad3antrrr 727 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58 simplr 766 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59 1st2nd 7940 . . . . . . . . . . . . . . . . 17 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6057, 58, 59syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
61 opeq1 4816 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6261adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6360, 62eqtr4d 2779 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (2nd𝑧)⟩)
64 difssd 4078 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
6564sselda 3931 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
6665adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧𝐹)
6763, 66eqeltrrd 2838 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)
6863, 67jca 512 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
69 opeq2 4817 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd𝑧)⟩)
7069eqeq2d 2747 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd𝑧)⟩))
7169eleq1d 2821 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
7270, 71anbi12d 631 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)))
7356, 68, 72spcedv 3546 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74 vex 3445 . . . . . . . . . . . . . 14 𝑥 ∈ V
7574elsnres 5957 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7673, 75sylibr 233 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
7713ad3antrrr 727 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝐹 Fn 𝐴)
7822ad2antrr 723 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑥𝐴)
79 fnressn 7080 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8077, 78, 79syl2anc 584 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8176, 80eleqtrd 2839 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩})
82 elsni 4589 . . . . . . . . . . 11 (𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8381, 82syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
84 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8584fveq2d 6823 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → (1st𝑧) = (1st ‘⟨𝑥, (𝐹𝑥)⟩))
86 fvex 6832 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
8774, 86op1st 7899 . . . . . . . . . . 11 (1st ‘⟨𝑥, (𝐹𝑥)⟩) = 𝑥
8885, 87eqtr2di 2793 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥 = (1st𝑧))
8983, 88impbida 798 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9089ralrimiva 3139 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9151, 55, 90rspcedvd 3572 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
92 reu6 3671 . . . . . . 7 (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
9391, 92sylibr 233 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧))
9417, 6, 7, 18, 8, 19, 20, 23, 40, 93gsummptf1o 19651 . . . . 5 (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
955, 16, 943eqtr3d 2784 . . . 4 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
96 simpr 485 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
9796eldifad 3909 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
98 funfv1st2nd 7947 . . . . . . 7 ((Fun 𝐹𝑧𝐹) → (𝐹‘(1st𝑧)) = (2nd𝑧))
9924, 97, 98syl2an2r 682 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st𝑧)) = (2nd𝑧))
10099mpteq2dva 5189 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧)))
101100oveq2d 7345 . . . 4 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
10295, 101eqtrd 2776 . . 3 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
103 nfcv 2904 . . . 4 𝑧(1st𝑡)
104 fvex 6832 . . . . 5 (2nd𝑡) ∈ V
105 fvex 6832 . . . . 5 (1st𝑡) ∈ V
106104, 105op2ndd 7902 . . . 4 (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ → (2nd𝑧) = (1st𝑡))
107 resfnfinfin 9189 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10813, 19, 107syl2anc 584 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10933, 108eqeltrrd 2838 . . . 4 (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
11033rneqd 5873 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111 rnresss 5953 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
1121frnd 6653 . . . . . 6 (𝜑 → ran 𝐹𝐵)
113111, 112sstrid 3942 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114110, 113eqsstrrd 3970 . . . 4 (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115 2ndrn 7942 . . . . 5 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
11627, 115sylan 580 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117 relcnv 6036 . . . . . . . 8 Rel 𝐹
118 reldif 5751 . . . . . . . 8 (Rel 𝐹 → Rel (𝐹 ∖ ({ 0 } × V)))
119117, 118mp1i 13 . . . . . . 7 (𝜑 → Rel (𝐹 ∖ ({ 0 } × V)))
120 1st2nd 7940 . . . . . . 7 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
121119, 120sylan 580 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
122 cnvdif 6076 . . . . . . . . . 10 (𝐹 ∖ (V × { 0 })) = (𝐹(V × { 0 }))
123 cnvxp 6089 . . . . . . . . . . 11 (V × { 0 }) = ({ 0 } × V)
124123difeq2i 4065 . . . . . . . . . 10 (𝐹(V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
125122, 124eqtri 2764 . . . . . . . . 9 (𝐹 ∖ (V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
126125eqimss2i 3990 . . . . . . . 8 (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 }))
127126a1i 11 . . . . . . 7 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 })))
128127sselda 3931 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡(𝐹 ∖ (V × { 0 })))
129121, 128eqeltrrd 2838 . . . . 5 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
130105, 104opelcnv 5817 . . . . 5 (⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131129, 130sylib 217 . . . 4 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
13227adantr 481 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133 eqidd 2737 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} = {𝑧})
134 cnvf1olem 8010 . . . . . . . . 9 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → ( {𝑧} ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑧 = { {𝑧}}))
135134simpld 495 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
136132, 96, 133, 135syl12anc 834 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
137136, 125eleqtrdi 2847 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ ({ 0 } × V)))
138 eqeq2 2748 . . . . . . . . 9 (𝑢 = {𝑧} → (𝑡 = 𝑢𝑡 = {𝑧}))
139138bibi2d 342 . . . . . . . 8 (𝑢 = {𝑧} → ((𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
140139ralbidv 3170 . . . . . . 7 (𝑢 = {𝑧} → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
141140adantl 482 . . . . . 6 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = {𝑧}) → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
142117, 118mp1i 13 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → Rel (𝐹 ∖ ({ 0 } × V)))
143 simplr 766 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
144 simpr 485 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
145 df-rel 5621 . . . . . . . . . . . . . 14 (Rel (𝐹 ∖ ({ 0 } × V)) ↔ (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146119, 145sylib 217 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147146ad3antrrr 727 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148147, 143sseldd 3932 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (V × V))
149 2nd1st 7939 . . . . . . . . . . 11 (𝑡 ∈ (V × V) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
150148, 149syl 17 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
151144, 150eqtr4d 2779 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = {𝑡})
152 cnvf1olem 8010 . . . . . . . . . 10 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → (𝑧(𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = {𝑧}))
153152simprd 496 . . . . . . . . 9 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → 𝑡 = {𝑧})
154142, 143, 151, 153syl12anc 834 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 = {𝑧})
15527ad3antrrr 727 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
15696ad2antrr 723 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157 simpr 485 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 = {𝑧})
158 cnvf1olem 8010 . . . . . . . . . . 11 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → (𝑡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = {𝑡}))
159158simprd 496 . . . . . . . . . 10 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → 𝑧 = {𝑡})
160155, 156, 157, 159syl12anc 834 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = {𝑡})
161146ad3antrrr 727 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162 simplr 766 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
163161, 162sseldd 3932 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (V × V))
164163, 149syl 17 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
165160, 164eqtrd 2776 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
166154, 165impbida 798 . . . . . . 7 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
167166ralrimiva 3139 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
168137, 141, 167rspcedvd 3572 . . . . 5 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
169 reu6 3671 . . . . 5 (∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
170168, 169sylibr 233 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
171103, 6, 7, 106, 8, 109, 114, 116, 131, 170gsummptf1o 19651 . . 3 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))) = (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))))
172 fveq2 6819 . . . . . 6 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
173172cbvmptv 5202 . . . . 5 (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧))
17433cnveqd 5811 . . . . . . 7 (𝜑(𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
175174, 125eqtr2di 2793 . . . . . 6 (𝜑 → (𝐹 ∖ ({ 0 } × V)) = (𝐹 ↾ (𝐹 supp 0 )))
176175mpteq1d 5184 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
177173, 176eqtrid 2788 . . . 4 (𝜑 → (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
178177oveq2d 7345 . . 3 (𝜑 → (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
179102, 171, 1783eqtrd 2780 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
180 nfcv 2904 . . 3 𝑦(1st𝑧)
181 nfv 1916 . . 3 𝑥𝜑
182 vex 3445 . . . 4 𝑦 ∈ V
18374, 182op1std 7901 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
184 relcnv 6036 . . . 4 Rel (𝐹 ↾ (𝐹 supp 0 ))
185184a1i 11 . . 3 (𝜑 → Rel (𝐹 ↾ (𝐹 supp 0 )))
186 cnvfi 9037 . . . 4 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187108, 186syl 17 . . 3 (𝜑(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188112adantr 481 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹𝐵)
189184a1i 11 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → Rel (𝐹 ↾ (𝐹 supp 0 )))
190 simpr 485 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧(𝐹 ↾ (𝐹 supp 0 )))
191 1stdm 7941 . . . . . . 7 ((Rel (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
192189, 190, 191syl2anc 584 . . . . . 6 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
193 df-rn 5625 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ↾ (𝐹 supp 0 ))
194192, 193eleqtrrdi 2848 . . . . 5 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195111, 194sselid 3929 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran 𝐹)
196188, 195sseldd 3932 . . 3 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ 𝐵)
197180, 181, 6, 183, 185, 187, 8, 196gsummpt2d 31509 . 2 (𝜑 → (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))) = (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198 df-ima 5627 . . . . . . 7 (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199 supppreima 31225 . . . . . . . . 9 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
20024, 12, 31, 199syl3anc 1370 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
201200imaeq2d 5993 . . . . . . 7 (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
202198, 201eqtr3id 2790 . . . . . 6 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
203 funimacnv 6559 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
20424, 203syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205 difssd 4078 . . . . . . 7 (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206 df-ss 3914 . . . . . . 7 ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207205, 206sylib 217 . . . . . 6 (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208202, 204, 2073eqtrd 2780 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209193, 208eqtr3id 2790 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
2108cmnmndd 19496 . . . . . . 7 (𝜑𝐺 ∈ Mnd)
211210adantr 481 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212108adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213 imafi2 31246 . . . . . . 7 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214212, 186, 2133syl 18 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215193, 113eqsstrrid 3980 . . . . . . 7 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216215sselda 3931 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥𝐵)
217 gsumhashmul.x . . . . . . 7 · = (.g𝐺)
2186, 217gsumconst 19622 . . . . . 6 ((𝐺 ∈ Mnd ∧ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥𝐵) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219211, 214, 216, 218syl3anc 1370 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220 cnvresima 6162 . . . . . . . 8 ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221209eleq2d 2822 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222221biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223222snssd 4755 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224 sspreima 6995 . . . . . . . . . . 11 ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
22524, 223, 224syl2an2r 682 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
226200adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
227225, 226sseqtrrd 3972 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228 df-ss 3914 . . . . . . . . 9 ((𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
229227, 228sylib 217 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
230220, 229eqtr2id 2789 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) = ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231230fveq2d 6823 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(𝐹 “ {𝑥})) = (♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232231oveq1d 7344 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(𝐹 “ {𝑥})) · 𝑥) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233219, 232eqtr4d 2779 . . . 4 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(𝐹 “ {𝑥})) · 𝑥))
234209, 233mpteq12dva 5178 . . 3 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥)))
235234oveq2d 7345 . 2 (𝜑 → (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
236179, 197, 2353eqtrd 2780 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wex 1780  wcel 2105  wral 3061  wrex 3070  ∃!wreu 3347  Vcvv 3441  cdif 3894  cin 3896  wss 3897  {csn 4572  cop 4578   cuni 4851   class class class wbr 5089  cmpt 5172   × cxp 5612  ccnv 5613  dom cdm 5614  ran crn 5615  cres 5616  cima 5617  Rel wrel 5619  Fun wfun 6467   Fn wfn 6468  wf 6469  cfv 6473  (class class class)co 7329  1st c1st 7889  2nd c2nd 7890   supp csupp 8039  Fincfn 8796   finSupp cfsupp 9218  chash 14137  Basecbs 17001  0gc0g 17239   Σg cgsu 17240  Mndcmnd 18474  .gcmg 18788  CMndccmn 19473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-of 7587  df-om 7773  df-1st 7891  df-2nd 7892  df-supp 8040  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-fsupp 9219  df-oi 9359  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476  df-seq 13815  df-hash 14138  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-0g 17241  df-gsum 17242  df-mre 17384  df-mrc 17385  df-acs 17387  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-submnd 18520  df-mulg 18789  df-cntz 19011  df-cmn 19475
This theorem is referenced by:  elrspunidl  31816
  Copyright terms: Public domain W3C validator